`UK Nonlinear News`,
`February 2001`

This was a 3 days meeting which was organized by the Dynamics Research Group in the Mathematics Department of the University of Warwick, UK from 19/1/2001 to 21/1/2001. The meeting was organized by A. Epstein, G. Havard, O. Sarig, and S. Van-Strien from Warwick, and was sponsored by the European Scientific Foundation as part of the PRODYN program.

**
List of Talks and Abstracts: **

Prof. Jon Aaronson

**Sums without Maxima (joint work with H. Nakada) **

Sums of positive random variables with "barely infinite"
expectations (excluding maximal values) sometimes enjoy a SLLN. There is
an application to some non uniformly hyperbolic ("intermittent")
interval
maps (joint work with Hitoshi Nakada).

Nalini Anantharaman

**Counting geodesics optimal in homology**

Artur Avila de Melo

**Statistical properties of unimodal maps
(Joint work with Carlos Gustavo Moreira) **

We discuss recent developments on the description of the
quadratic family and other families of unimodal maps from
the measure-theoretical point of
view. In particular, we show that typical unimodal maps are either
hyperbolic or Collet-Eckmann, with polynomial recurrence of the critical
orbit. In both cases the dynamics is very nice from the statistical point
of view.

Viviane Baladi

**Floquet spectrum of weakly coupled analytic maps
(Joint work with HH Rugh) **

We consider translation invariant
weakly coupled analytic expanding circle
maps on Z^D, with small coupling strength
and study the spectrum of the associated transfer
operator restricted to the eigenspaces of the
spatial translation operators. (The operators commute.)
We exhibit smooth curves of eigenspaces and
eigenfunctions (as functions of the crystal
momenta of the spatial translations).

Xavier Bressaud

**Expanding interval maps with intermittent-like behaviour: Physical measures and scales of
time**

Arnaud Cheritat

**Virtual Siegel disks**

Local connectivity and measure zero results for Julia sets
of some maps z -> rho.z + z^2 having a Siegel disk at 0, have been proved
with the use of a model, a Blaschke fraction, usually given by a formula.
This fraction can also be constructed in a geometric way. By analogy, one
can construct geometrically some models (for which one has no formula),
that enable to adapt the results to virtual Siegel disks of Julia-Lavaurs
sets that appear in parabolic implosion.

Zaqueu Coelho

**The loss of tightness of time distributions for rotations**

For an irrational rotation of the circle, the distribution
of times to hit a small interval (when rescaled by the size of the
interval) may lose tightness when the interval shrinks to a point.

Sebastien Ferenczi

**Structure of three-interval exchange transformations (joint work with C. Holton and L. Zamboni)**

We present a detailed study of the combinatorial,
spectral and ergodic properties of three-interval exchange transformations.
We give first a necessary and
sufficient condition
for a symbolic system to be a natural coding of a three-interval exchange,
answering an old question of Rauzy; this allows us to define
a new algorithm of simultaneous approximations for two real numbers,
generalizing the well-known interaction between rotations, Sturmian sequences
and the usual continued fraction approximation.
Then we define a recursive method of
generating three sequences of nested Rokhlin stacks which describe the system
from a measure-theoretic point of view and which in turn gives an explicit
characterization of the eigenvalues. We obtain necessary and sufficient
conditions for weak mixing which, in addition to unifying all previously
known examples, allow us to exhibit new interesting examples of weakly
mixing three-interval exchanges. Finally we give affirmative answers to two
long standing questions posed by W.A. Veech on the existence of
three-interval
exchanges having irrational eigenvalues and discrete spectrum.

Oliver Jenkinson

**Cohomology classes for Dynamically Non-Negative Functions"
(joint work with T. Bousch) **

Given a dynamical system $T:X \to X$, we say that a continuous
function $f:X\to R$ is dynamically non-negative (resp. dynamically
positive) if $\int f dm \ge 0$ (resp. $\int f dm > 0$ ) for every
$T$-invariant probability measure $m$.
Given such a function $f$, can we always find a bona fide
non-negative (resp. positive) function $g$ in the same
dynamic cohomology class? (i.e. $f-g= uT - u$ for some continuous $u$).
If $f$ has some regularity (eg Holder, $C^k$, analytic, etc),
can we choose this $g$ to have the same regularity?
We consider these problems in the case where $T$ is an expanding map.

Marc Kesseboehmer

**Poisson limit laws for the Gauss map**

We introduce a general theorem which states sufficient conditions for
schemes of dependent random variables to fulfill a Poisson limit law.
Also explicit bounds for the error terms are provided. We then apply
this theorem to the return process for the (measure theoretical)
dynamical system given by the Gauss map $\sigma$: For almost every
$x\in [0,1]$ we have
\[
\mu \{x\in [0,1]: \sum _{i=1}^{[\lambda / \mu(B_r (x))]} \chi
_{B_r(x)}\circ \sigma ^i (x)=k\}\to \exp(-\lambda)\lambda^k /k\! \]
with exponential rate ($\mu$ denotes the Gauss measure).

Makoto Mori

**Low discrepancy sequences generated by piecewise linear
transformations**

We construct a uniformly distributed sequences by expanding
maps, and characterizes discrepancy of sequences by the spectra of the
Perron-Frobenius operator.

Duncan Sands

**Metric attractors for smooth unimodal maps
(joint work with J. Graczyk & G. Swiatek)**

We prove decay of geometry for $C^3$ unimodal maps with $C^3$ non-flat
critical points. As a consequence we obtain the classification of the
measure theoretical attractors for $C^3$ unimodal maps with non-flat
critical points of order 2.

Richard Sharp

**A Local limit theorem for closed geodesic and homology**

We discuss the distribution of closed geodesics on a negatively
curved manifold. We study the asymptotics of the number of closed
geodesics lying in a prescribed homology class. Under certain
conditions, we obtain a local limit theorem which gives information
which is uniform as the homology class varies.

Michel Zinsmeister

**Continuity with respect to the phase of Hausdorff dimension of
Julia-Lavaurs sets**

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Page Created: 18th January 2001.

Last Updated: 5th February 2001.